Problem: What is the extraneous solution to these equations? $\dfrac{x^2}{x - 4} = \dfrac{x + 12}{x - 4}$
Solution: Multiply both sides by $x - 4$ $ \dfrac{x^2}{x - 4} (x - 4) = \dfrac{x + 12}{x - 4} (x - 4)$ $ x^2 = x + 12$ Subtract $x + 12$ from both sides: $ x^2 - (x + 12) = x + 12 - (x + 12)$ $ x^2 - x - 12 = 0$ Factor the expression: $ (x - 4)(x + 3) = 0$ Therefore $x = 4$ or $x = -3$ At $x = 4$ , the denominator of the original expression is 0. Since the expression is undefined at $x = 4$, it is an extraneous solution.